Properties

Label 8470.2.a.dc.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.19898000.1
Defining polynomial: \(x^{6} - x^{5} - 10 x^{4} + 7 x^{3} + 24 x^{2} - 15 x - 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.86564\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.86564 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.86564 q^{6} -1.00000 q^{7} +1.00000 q^{8} +5.21187 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.86564 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.86564 q^{6} -1.00000 q^{7} +1.00000 q^{8} +5.21187 q^{9} +1.00000 q^{10} -2.86564 q^{12} +5.63670 q^{13} -1.00000 q^{14} -2.86564 q^{15} +1.00000 q^{16} +2.32986 q^{17} +5.21187 q^{18} +7.81494 q^{19} +1.00000 q^{20} +2.86564 q^{21} +7.98434 q^{23} -2.86564 q^{24} +1.00000 q^{25} +5.63670 q^{26} -6.33840 q^{27} -1.00000 q^{28} -2.99446 q^{29} -2.86564 q^{30} +5.08533 q^{31} +1.00000 q^{32} +2.32986 q^{34} -1.00000 q^{35} +5.21187 q^{36} -1.38681 q^{37} +7.81494 q^{38} -16.1527 q^{39} +1.00000 q^{40} -6.90154 q^{41} +2.86564 q^{42} +11.5492 q^{43} +5.21187 q^{45} +7.98434 q^{46} -5.14520 q^{47} -2.86564 q^{48} +1.00000 q^{49} +1.00000 q^{50} -6.67652 q^{51} +5.63670 q^{52} +0.582695 q^{53} -6.33840 q^{54} -1.00000 q^{56} -22.3948 q^{57} -2.99446 q^{58} +13.8292 q^{59} -2.86564 q^{60} -7.82443 q^{61} +5.08533 q^{62} -5.21187 q^{63} +1.00000 q^{64} +5.63670 q^{65} -0.0485769 q^{67} +2.32986 q^{68} -22.8802 q^{69} -1.00000 q^{70} +4.45361 q^{71} +5.21187 q^{72} +0.336814 q^{73} -1.38681 q^{74} -2.86564 q^{75} +7.81494 q^{76} -16.1527 q^{78} -1.05291 q^{79} +1.00000 q^{80} +2.52795 q^{81} -6.90154 q^{82} -10.8315 q^{83} +2.86564 q^{84} +2.32986 q^{85} +11.5492 q^{86} +8.58103 q^{87} -18.5268 q^{89} +5.21187 q^{90} -5.63670 q^{91} +7.98434 q^{92} -14.5727 q^{93} -5.14520 q^{94} +7.81494 q^{95} -2.86564 q^{96} -5.33180 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} + 3 q^{9} + O(q^{10}) \) \( 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} + 3 q^{9} + 6 q^{10} - q^{12} + 9 q^{13} - 6 q^{14} - q^{15} + 6 q^{16} + 9 q^{17} + 3 q^{18} + 12 q^{19} + 6 q^{20} + q^{21} + 4 q^{23} - q^{24} + 6 q^{25} + 9 q^{26} - 4 q^{27} - 6 q^{28} + 15 q^{29} - q^{30} + 8 q^{31} + 6 q^{32} + 9 q^{34} - 6 q^{35} + 3 q^{36} - 4 q^{37} + 12 q^{38} - 19 q^{39} + 6 q^{40} + 4 q^{41} + q^{42} + 30 q^{43} + 3 q^{45} + 4 q^{46} - 7 q^{47} - q^{48} + 6 q^{49} + 6 q^{50} + 16 q^{51} + 9 q^{52} - 6 q^{53} - 4 q^{54} - 6 q^{56} - 14 q^{57} + 15 q^{58} + 4 q^{59} - q^{60} - 14 q^{61} + 8 q^{62} - 3 q^{63} + 6 q^{64} + 9 q^{65} + 18 q^{67} + 9 q^{68} - 10 q^{69} - 6 q^{70} + 23 q^{71} + 3 q^{72} + 23 q^{73} - 4 q^{74} - q^{75} + 12 q^{76} - 19 q^{78} + 21 q^{79} + 6 q^{80} - 18 q^{81} + 4 q^{82} + 25 q^{83} + q^{84} + 9 q^{85} + 30 q^{86} + 14 q^{87} - 18 q^{89} + 3 q^{90} - 9 q^{91} + 4 q^{92} - 24 q^{93} - 7 q^{94} + 12 q^{95} - q^{96} + 7 q^{97} + 6 q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −2.86564 −1.65448 −0.827238 0.561852i \(-0.810089\pi\)
−0.827238 + 0.561852i \(0.810089\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −2.86564 −1.16989
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 5.21187 1.73729
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) −2.86564 −0.827238
\(13\) 5.63670 1.56334 0.781669 0.623693i \(-0.214369\pi\)
0.781669 + 0.623693i \(0.214369\pi\)
\(14\) −1.00000 −0.267261
\(15\) −2.86564 −0.739904
\(16\) 1.00000 0.250000
\(17\) 2.32986 0.565074 0.282537 0.959256i \(-0.408824\pi\)
0.282537 + 0.959256i \(0.408824\pi\)
\(18\) 5.21187 1.22845
\(19\) 7.81494 1.79287 0.896435 0.443175i \(-0.146148\pi\)
0.896435 + 0.443175i \(0.146148\pi\)
\(20\) 1.00000 0.223607
\(21\) 2.86564 0.625333
\(22\) 0 0
\(23\) 7.98434 1.66485 0.832425 0.554138i \(-0.186952\pi\)
0.832425 + 0.554138i \(0.186952\pi\)
\(24\) −2.86564 −0.584945
\(25\) 1.00000 0.200000
\(26\) 5.63670 1.10545
\(27\) −6.33840 −1.21983
\(28\) −1.00000 −0.188982
\(29\) −2.99446 −0.556057 −0.278029 0.960573i \(-0.589681\pi\)
−0.278029 + 0.960573i \(0.589681\pi\)
\(30\) −2.86564 −0.523191
\(31\) 5.08533 0.913352 0.456676 0.889633i \(-0.349040\pi\)
0.456676 + 0.889633i \(0.349040\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.32986 0.399567
\(35\) −1.00000 −0.169031
\(36\) 5.21187 0.868644
\(37\) −1.38681 −0.227989 −0.113995 0.993481i \(-0.536365\pi\)
−0.113995 + 0.993481i \(0.536365\pi\)
\(38\) 7.81494 1.26775
\(39\) −16.1527 −2.58650
\(40\) 1.00000 0.158114
\(41\) −6.90154 −1.07784 −0.538920 0.842357i \(-0.681167\pi\)
−0.538920 + 0.842357i \(0.681167\pi\)
\(42\) 2.86564 0.442177
\(43\) 11.5492 1.76124 0.880622 0.473820i \(-0.157125\pi\)
0.880622 + 0.473820i \(0.157125\pi\)
\(44\) 0 0
\(45\) 5.21187 0.776939
\(46\) 7.98434 1.17723
\(47\) −5.14520 −0.750504 −0.375252 0.926923i \(-0.622444\pi\)
−0.375252 + 0.926923i \(0.622444\pi\)
\(48\) −2.86564 −0.413619
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −6.67652 −0.934900
\(52\) 5.63670 0.781669
\(53\) 0.582695 0.0800393 0.0400196 0.999199i \(-0.487258\pi\)
0.0400196 + 0.999199i \(0.487258\pi\)
\(54\) −6.33840 −0.862547
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −22.3948 −2.96626
\(58\) −2.99446 −0.393192
\(59\) 13.8292 1.80041 0.900204 0.435468i \(-0.143417\pi\)
0.900204 + 0.435468i \(0.143417\pi\)
\(60\) −2.86564 −0.369952
\(61\) −7.82443 −1.00182 −0.500908 0.865501i \(-0.667000\pi\)
−0.500908 + 0.865501i \(0.667000\pi\)
\(62\) 5.08533 0.645838
\(63\) −5.21187 −0.656633
\(64\) 1.00000 0.125000
\(65\) 5.63670 0.699146
\(66\) 0 0
\(67\) −0.0485769 −0.00593462 −0.00296731 0.999996i \(-0.500945\pi\)
−0.00296731 + 0.999996i \(0.500945\pi\)
\(68\) 2.32986 0.282537
\(69\) −22.8802 −2.75445
\(70\) −1.00000 −0.119523
\(71\) 4.45361 0.528546 0.264273 0.964448i \(-0.414868\pi\)
0.264273 + 0.964448i \(0.414868\pi\)
\(72\) 5.21187 0.614224
\(73\) 0.336814 0.0394210 0.0197105 0.999806i \(-0.493726\pi\)
0.0197105 + 0.999806i \(0.493726\pi\)
\(74\) −1.38681 −0.161213
\(75\) −2.86564 −0.330895
\(76\) 7.81494 0.896435
\(77\) 0 0
\(78\) −16.1527 −1.82893
\(79\) −1.05291 −0.118462 −0.0592308 0.998244i \(-0.518865\pi\)
−0.0592308 + 0.998244i \(0.518865\pi\)
\(80\) 1.00000 0.111803
\(81\) 2.52795 0.280883
\(82\) −6.90154 −0.762148
\(83\) −10.8315 −1.18891 −0.594455 0.804129i \(-0.702632\pi\)
−0.594455 + 0.804129i \(0.702632\pi\)
\(84\) 2.86564 0.312666
\(85\) 2.32986 0.252709
\(86\) 11.5492 1.24539
\(87\) 8.58103 0.919983
\(88\) 0 0
\(89\) −18.5268 −1.96384 −0.981921 0.189291i \(-0.939381\pi\)
−0.981921 + 0.189291i \(0.939381\pi\)
\(90\) 5.21187 0.549379
\(91\) −5.63670 −0.590886
\(92\) 7.98434 0.832425
\(93\) −14.5727 −1.51112
\(94\) −5.14520 −0.530686
\(95\) 7.81494 0.801796
\(96\) −2.86564 −0.292473
\(97\) −5.33180 −0.541362 −0.270681 0.962669i \(-0.587249\pi\)
−0.270681 + 0.962669i \(0.587249\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 18.2701 1.81795 0.908973 0.416854i \(-0.136867\pi\)
0.908973 + 0.416854i \(0.136867\pi\)
\(102\) −6.67652 −0.661074
\(103\) −11.2347 −1.10699 −0.553495 0.832852i \(-0.686706\pi\)
−0.553495 + 0.832852i \(0.686706\pi\)
\(104\) 5.63670 0.552723
\(105\) 2.86564 0.279657
\(106\) 0.582695 0.0565963
\(107\) −4.89157 −0.472886 −0.236443 0.971645i \(-0.575982\pi\)
−0.236443 + 0.971645i \(0.575982\pi\)
\(108\) −6.33840 −0.609913
\(109\) −6.26442 −0.600023 −0.300011 0.953936i \(-0.596991\pi\)
−0.300011 + 0.953936i \(0.596991\pi\)
\(110\) 0 0
\(111\) 3.97408 0.377203
\(112\) −1.00000 −0.0944911
\(113\) 10.1651 0.956251 0.478126 0.878291i \(-0.341316\pi\)
0.478126 + 0.878291i \(0.341316\pi\)
\(114\) −22.3948 −2.09746
\(115\) 7.98434 0.744544
\(116\) −2.99446 −0.278029
\(117\) 29.3777 2.71597
\(118\) 13.8292 1.27308
\(119\) −2.32986 −0.213578
\(120\) −2.86564 −0.261596
\(121\) 0 0
\(122\) −7.82443 −0.708391
\(123\) 19.7773 1.78326
\(124\) 5.08533 0.456676
\(125\) 1.00000 0.0894427
\(126\) −5.21187 −0.464310
\(127\) −6.58727 −0.584526 −0.292263 0.956338i \(-0.594408\pi\)
−0.292263 + 0.956338i \(0.594408\pi\)
\(128\) 1.00000 0.0883883
\(129\) −33.0959 −2.91393
\(130\) 5.63670 0.494371
\(131\) −1.20821 −0.105562 −0.0527808 0.998606i \(-0.516808\pi\)
−0.0527808 + 0.998606i \(0.516808\pi\)
\(132\) 0 0
\(133\) −7.81494 −0.677641
\(134\) −0.0485769 −0.00419641
\(135\) −6.33840 −0.545523
\(136\) 2.32986 0.199784
\(137\) 14.0829 1.20318 0.601590 0.798805i \(-0.294534\pi\)
0.601590 + 0.798805i \(0.294534\pi\)
\(138\) −22.8802 −1.94769
\(139\) −2.74290 −0.232650 −0.116325 0.993211i \(-0.537111\pi\)
−0.116325 + 0.993211i \(0.537111\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 14.7443 1.24169
\(142\) 4.45361 0.373739
\(143\) 0 0
\(144\) 5.21187 0.434322
\(145\) −2.99446 −0.248676
\(146\) 0.336814 0.0278749
\(147\) −2.86564 −0.236354
\(148\) −1.38681 −0.113995
\(149\) 17.1642 1.40615 0.703075 0.711116i \(-0.251810\pi\)
0.703075 + 0.711116i \(0.251810\pi\)
\(150\) −2.86564 −0.233978
\(151\) −5.94304 −0.483637 −0.241819 0.970321i \(-0.577744\pi\)
−0.241819 + 0.970321i \(0.577744\pi\)
\(152\) 7.81494 0.633875
\(153\) 12.1429 0.981696
\(154\) 0 0
\(155\) 5.08533 0.408464
\(156\) −16.1527 −1.29325
\(157\) −16.7947 −1.34037 −0.670183 0.742196i \(-0.733784\pi\)
−0.670183 + 0.742196i \(0.733784\pi\)
\(158\) −1.05291 −0.0837650
\(159\) −1.66979 −0.132423
\(160\) 1.00000 0.0790569
\(161\) −7.98434 −0.629254
\(162\) 2.52795 0.198614
\(163\) 2.43041 0.190364 0.0951821 0.995460i \(-0.469657\pi\)
0.0951821 + 0.995460i \(0.469657\pi\)
\(164\) −6.90154 −0.538920
\(165\) 0 0
\(166\) −10.8315 −0.840686
\(167\) 11.0306 0.853572 0.426786 0.904353i \(-0.359646\pi\)
0.426786 + 0.904353i \(0.359646\pi\)
\(168\) 2.86564 0.221089
\(169\) 18.7723 1.44403
\(170\) 2.32986 0.178692
\(171\) 40.7304 3.11473
\(172\) 11.5492 0.880622
\(173\) −14.6947 −1.11721 −0.558607 0.829433i \(-0.688664\pi\)
−0.558607 + 0.829433i \(0.688664\pi\)
\(174\) 8.58103 0.650526
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −39.6294 −2.97873
\(178\) −18.5268 −1.38865
\(179\) 4.51392 0.337387 0.168693 0.985669i \(-0.446045\pi\)
0.168693 + 0.985669i \(0.446045\pi\)
\(180\) 5.21187 0.388470
\(181\) −10.1770 −0.756451 −0.378225 0.925714i \(-0.623466\pi\)
−0.378225 + 0.925714i \(0.623466\pi\)
\(182\) −5.63670 −0.417820
\(183\) 22.4220 1.65748
\(184\) 7.98434 0.588613
\(185\) −1.38681 −0.101960
\(186\) −14.5727 −1.06852
\(187\) 0 0
\(188\) −5.14520 −0.375252
\(189\) 6.33840 0.461051
\(190\) 7.81494 0.566955
\(191\) 11.9914 0.867669 0.433834 0.900993i \(-0.357160\pi\)
0.433834 + 0.900993i \(0.357160\pi\)
\(192\) −2.86564 −0.206809
\(193\) 17.8748 1.28666 0.643330 0.765589i \(-0.277552\pi\)
0.643330 + 0.765589i \(0.277552\pi\)
\(194\) −5.33180 −0.382801
\(195\) −16.1527 −1.15672
\(196\) 1.00000 0.0714286
\(197\) 11.7871 0.839795 0.419897 0.907572i \(-0.362066\pi\)
0.419897 + 0.907572i \(0.362066\pi\)
\(198\) 0 0
\(199\) −13.9342 −0.987767 −0.493884 0.869528i \(-0.664423\pi\)
−0.493884 + 0.869528i \(0.664423\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0.139204 0.00981868
\(202\) 18.2701 1.28548
\(203\) 2.99446 0.210170
\(204\) −6.67652 −0.467450
\(205\) −6.90154 −0.482025
\(206\) −11.2347 −0.782760
\(207\) 41.6133 2.89233
\(208\) 5.63670 0.390835
\(209\) 0 0
\(210\) 2.86564 0.197748
\(211\) −26.4497 −1.82087 −0.910435 0.413653i \(-0.864253\pi\)
−0.910435 + 0.413653i \(0.864253\pi\)
\(212\) 0.582695 0.0400196
\(213\) −12.7624 −0.874467
\(214\) −4.89157 −0.334381
\(215\) 11.5492 0.787652
\(216\) −6.33840 −0.431274
\(217\) −5.08533 −0.345215
\(218\) −6.26442 −0.424280
\(219\) −0.965185 −0.0652211
\(220\) 0 0
\(221\) 13.1327 0.883401
\(222\) 3.97408 0.266723
\(223\) 24.5893 1.64662 0.823311 0.567591i \(-0.192124\pi\)
0.823311 + 0.567591i \(0.192124\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 5.21187 0.347458
\(226\) 10.1651 0.676172
\(227\) 5.59242 0.371182 0.185591 0.982627i \(-0.440580\pi\)
0.185591 + 0.982627i \(0.440580\pi\)
\(228\) −22.3948 −1.48313
\(229\) −8.10340 −0.535488 −0.267744 0.963490i \(-0.586278\pi\)
−0.267744 + 0.963490i \(0.586278\pi\)
\(230\) 7.98434 0.526472
\(231\) 0 0
\(232\) −2.99446 −0.196596
\(233\) −11.9807 −0.784882 −0.392441 0.919777i \(-0.628369\pi\)
−0.392441 + 0.919777i \(0.628369\pi\)
\(234\) 29.3777 1.92048
\(235\) −5.14520 −0.335636
\(236\) 13.8292 0.900204
\(237\) 3.01726 0.195992
\(238\) −2.32986 −0.151022
\(239\) 6.46612 0.418258 0.209129 0.977888i \(-0.432937\pi\)
0.209129 + 0.977888i \(0.432937\pi\)
\(240\) −2.86564 −0.184976
\(241\) −20.3441 −1.31048 −0.655238 0.755423i \(-0.727432\pi\)
−0.655238 + 0.755423i \(0.727432\pi\)
\(242\) 0 0
\(243\) 11.7710 0.755112
\(244\) −7.82443 −0.500908
\(245\) 1.00000 0.0638877
\(246\) 19.7773 1.26096
\(247\) 44.0504 2.80286
\(248\) 5.08533 0.322919
\(249\) 31.0391 1.96702
\(250\) 1.00000 0.0632456
\(251\) 12.5842 0.794311 0.397155 0.917751i \(-0.369997\pi\)
0.397155 + 0.917751i \(0.369997\pi\)
\(252\) −5.21187 −0.328317
\(253\) 0 0
\(254\) −6.58727 −0.413322
\(255\) −6.67652 −0.418100
\(256\) 1.00000 0.0625000
\(257\) 3.45995 0.215826 0.107913 0.994160i \(-0.465583\pi\)
0.107913 + 0.994160i \(0.465583\pi\)
\(258\) −33.0959 −2.06046
\(259\) 1.38681 0.0861719
\(260\) 5.63670 0.349573
\(261\) −15.6067 −0.966032
\(262\) −1.20821 −0.0746434
\(263\) 8.01789 0.494404 0.247202 0.968964i \(-0.420489\pi\)
0.247202 + 0.968964i \(0.420489\pi\)
\(264\) 0 0
\(265\) 0.582695 0.0357946
\(266\) −7.81494 −0.479165
\(267\) 53.0912 3.24913
\(268\) −0.0485769 −0.00296731
\(269\) −18.5650 −1.13193 −0.565964 0.824430i \(-0.691496\pi\)
−0.565964 + 0.824430i \(0.691496\pi\)
\(270\) −6.33840 −0.385743
\(271\) −25.2051 −1.53110 −0.765552 0.643375i \(-0.777534\pi\)
−0.765552 + 0.643375i \(0.777534\pi\)
\(272\) 2.32986 0.141268
\(273\) 16.1527 0.977607
\(274\) 14.0829 0.850777
\(275\) 0 0
\(276\) −22.8802 −1.37723
\(277\) 12.2281 0.734714 0.367357 0.930080i \(-0.380263\pi\)
0.367357 + 0.930080i \(0.380263\pi\)
\(278\) −2.74290 −0.164508
\(279\) 26.5041 1.58676
\(280\) −1.00000 −0.0597614
\(281\) −26.5934 −1.58643 −0.793216 0.608941i \(-0.791595\pi\)
−0.793216 + 0.608941i \(0.791595\pi\)
\(282\) 14.7443 0.878008
\(283\) 13.7220 0.815689 0.407845 0.913051i \(-0.366281\pi\)
0.407845 + 0.913051i \(0.366281\pi\)
\(284\) 4.45361 0.264273
\(285\) −22.3948 −1.32655
\(286\) 0 0
\(287\) 6.90154 0.407385
\(288\) 5.21187 0.307112
\(289\) −11.5718 −0.680692
\(290\) −2.99446 −0.175841
\(291\) 15.2790 0.895670
\(292\) 0.336814 0.0197105
\(293\) −0.714463 −0.0417394 −0.0208697 0.999782i \(-0.506644\pi\)
−0.0208697 + 0.999782i \(0.506644\pi\)
\(294\) −2.86564 −0.167127
\(295\) 13.8292 0.805167
\(296\) −1.38681 −0.0806064
\(297\) 0 0
\(298\) 17.1642 0.994298
\(299\) 45.0053 2.60272
\(300\) −2.86564 −0.165448
\(301\) −11.5492 −0.665688
\(302\) −5.94304 −0.341983
\(303\) −52.3556 −3.00775
\(304\) 7.81494 0.448218
\(305\) −7.82443 −0.448026
\(306\) 12.1429 0.694164
\(307\) 19.3442 1.10403 0.552015 0.833834i \(-0.313859\pi\)
0.552015 + 0.833834i \(0.313859\pi\)
\(308\) 0 0
\(309\) 32.1946 1.83149
\(310\) 5.08533 0.288827
\(311\) 1.82729 0.103616 0.0518080 0.998657i \(-0.483502\pi\)
0.0518080 + 0.998657i \(0.483502\pi\)
\(312\) −16.1527 −0.914467
\(313\) −9.83108 −0.555686 −0.277843 0.960627i \(-0.589619\pi\)
−0.277843 + 0.960627i \(0.589619\pi\)
\(314\) −16.7947 −0.947782
\(315\) −5.21187 −0.293655
\(316\) −1.05291 −0.0592308
\(317\) −13.8545 −0.778146 −0.389073 0.921207i \(-0.627205\pi\)
−0.389073 + 0.921207i \(0.627205\pi\)
\(318\) −1.66979 −0.0936372
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 14.0175 0.782378
\(322\) −7.98434 −0.444950
\(323\) 18.2077 1.01310
\(324\) 2.52795 0.140442
\(325\) 5.63670 0.312668
\(326\) 2.43041 0.134608
\(327\) 17.9516 0.992723
\(328\) −6.90154 −0.381074
\(329\) 5.14520 0.283664
\(330\) 0 0
\(331\) 8.02084 0.440865 0.220433 0.975402i \(-0.429253\pi\)
0.220433 + 0.975402i \(0.429253\pi\)
\(332\) −10.8315 −0.594455
\(333\) −7.22784 −0.396083
\(334\) 11.0306 0.603567
\(335\) −0.0485769 −0.00265404
\(336\) 2.86564 0.156333
\(337\) 11.6060 0.632218 0.316109 0.948723i \(-0.397624\pi\)
0.316109 + 0.948723i \(0.397624\pi\)
\(338\) 18.7723 1.02108
\(339\) −29.1294 −1.58209
\(340\) 2.32986 0.126354
\(341\) 0 0
\(342\) 40.7304 2.20245
\(343\) −1.00000 −0.0539949
\(344\) 11.5492 0.622694
\(345\) −22.8802 −1.23183
\(346\) −14.6947 −0.789989
\(347\) 26.4301 1.41884 0.709420 0.704786i \(-0.248957\pi\)
0.709420 + 0.704786i \(0.248957\pi\)
\(348\) 8.58103 0.459991
\(349\) −27.3777 −1.46549 −0.732747 0.680501i \(-0.761762\pi\)
−0.732747 + 0.680501i \(0.761762\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −35.7276 −1.90700
\(352\) 0 0
\(353\) 21.3772 1.13779 0.568897 0.822409i \(-0.307370\pi\)
0.568897 + 0.822409i \(0.307370\pi\)
\(354\) −39.6294 −2.10628
\(355\) 4.45361 0.236373
\(356\) −18.5268 −0.981921
\(357\) 6.67652 0.353359
\(358\) 4.51392 0.238568
\(359\) 10.8904 0.574774 0.287387 0.957815i \(-0.407213\pi\)
0.287387 + 0.957815i \(0.407213\pi\)
\(360\) 5.21187 0.274689
\(361\) 42.0733 2.21439
\(362\) −10.1770 −0.534891
\(363\) 0 0
\(364\) −5.63670 −0.295443
\(365\) 0.336814 0.0176296
\(366\) 22.4220 1.17201
\(367\) 11.3108 0.590416 0.295208 0.955433i \(-0.404611\pi\)
0.295208 + 0.955433i \(0.404611\pi\)
\(368\) 7.98434 0.416213
\(369\) −35.9699 −1.87252
\(370\) −1.38681 −0.0720966
\(371\) −0.582695 −0.0302520
\(372\) −14.5727 −0.755559
\(373\) −11.6658 −0.604034 −0.302017 0.953303i \(-0.597660\pi\)
−0.302017 + 0.953303i \(0.597660\pi\)
\(374\) 0 0
\(375\) −2.86564 −0.147981
\(376\) −5.14520 −0.265343
\(377\) −16.8789 −0.869305
\(378\) 6.33840 0.326012
\(379\) 0.617088 0.0316977 0.0158488 0.999874i \(-0.494955\pi\)
0.0158488 + 0.999874i \(0.494955\pi\)
\(380\) 7.81494 0.400898
\(381\) 18.8767 0.967084
\(382\) 11.9914 0.613535
\(383\) −26.4471 −1.35138 −0.675692 0.737184i \(-0.736155\pi\)
−0.675692 + 0.737184i \(0.736155\pi\)
\(384\) −2.86564 −0.146236
\(385\) 0 0
\(386\) 17.8748 0.909806
\(387\) 60.1931 3.05979
\(388\) −5.33180 −0.270681
\(389\) 1.60446 0.0813492 0.0406746 0.999172i \(-0.487049\pi\)
0.0406746 + 0.999172i \(0.487049\pi\)
\(390\) −16.1527 −0.817924
\(391\) 18.6024 0.940763
\(392\) 1.00000 0.0505076
\(393\) 3.46228 0.174649
\(394\) 11.7871 0.593825
\(395\) −1.05291 −0.0529776
\(396\) 0 0
\(397\) 5.28819 0.265407 0.132703 0.991156i \(-0.457634\pi\)
0.132703 + 0.991156i \(0.457634\pi\)
\(398\) −13.9342 −0.698457
\(399\) 22.3948 1.12114
\(400\) 1.00000 0.0500000
\(401\) −13.6059 −0.679446 −0.339723 0.940526i \(-0.610333\pi\)
−0.339723 + 0.940526i \(0.610333\pi\)
\(402\) 0.139204 0.00694285
\(403\) 28.6645 1.42788
\(404\) 18.2701 0.908973
\(405\) 2.52795 0.125615
\(406\) 2.99446 0.148613
\(407\) 0 0
\(408\) −6.67652 −0.330537
\(409\) 3.40230 0.168233 0.0841164 0.996456i \(-0.473193\pi\)
0.0841164 + 0.996456i \(0.473193\pi\)
\(410\) −6.90154 −0.340843
\(411\) −40.3564 −1.99063
\(412\) −11.2347 −0.553495
\(413\) −13.8292 −0.680490
\(414\) 41.6133 2.04518
\(415\) −10.8315 −0.531697
\(416\) 5.63670 0.276362
\(417\) 7.86016 0.384914
\(418\) 0 0
\(419\) −39.5263 −1.93099 −0.965493 0.260428i \(-0.916136\pi\)
−0.965493 + 0.260428i \(0.916136\pi\)
\(420\) 2.86564 0.139829
\(421\) −15.6617 −0.763304 −0.381652 0.924306i \(-0.624645\pi\)
−0.381652 + 0.924306i \(0.624645\pi\)
\(422\) −26.4497 −1.28755
\(423\) −26.8161 −1.30384
\(424\) 0.582695 0.0282982
\(425\) 2.32986 0.113015
\(426\) −12.7624 −0.618341
\(427\) 7.82443 0.378651
\(428\) −4.89157 −0.236443
\(429\) 0 0
\(430\) 11.5492 0.556954
\(431\) −5.63150 −0.271260 −0.135630 0.990760i \(-0.543306\pi\)
−0.135630 + 0.990760i \(0.543306\pi\)
\(432\) −6.33840 −0.304956
\(433\) −34.5223 −1.65903 −0.829517 0.558481i \(-0.811384\pi\)
−0.829517 + 0.558481i \(0.811384\pi\)
\(434\) −5.08533 −0.244104
\(435\) 8.58103 0.411429
\(436\) −6.26442 −0.300011
\(437\) 62.3972 2.98486
\(438\) −0.965185 −0.0461183
\(439\) −30.0533 −1.43437 −0.717184 0.696884i \(-0.754569\pi\)
−0.717184 + 0.696884i \(0.754569\pi\)
\(440\) 0 0
\(441\) 5.21187 0.248184
\(442\) 13.1327 0.624659
\(443\) −32.0651 −1.52346 −0.761730 0.647894i \(-0.775650\pi\)
−0.761730 + 0.647894i \(0.775650\pi\)
\(444\) 3.97408 0.188601
\(445\) −18.5268 −0.878257
\(446\) 24.5893 1.16434
\(447\) −49.1865 −2.32644
\(448\) −1.00000 −0.0472456
\(449\) −9.27706 −0.437811 −0.218906 0.975746i \(-0.570249\pi\)
−0.218906 + 0.975746i \(0.570249\pi\)
\(450\) 5.21187 0.245690
\(451\) 0 0
\(452\) 10.1651 0.478126
\(453\) 17.0306 0.800166
\(454\) 5.59242 0.262465
\(455\) −5.63670 −0.264252
\(456\) −22.3948 −1.04873
\(457\) 20.0782 0.939217 0.469608 0.882875i \(-0.344395\pi\)
0.469608 + 0.882875i \(0.344395\pi\)
\(458\) −8.10340 −0.378647
\(459\) −14.7676 −0.689291
\(460\) 7.98434 0.372272
\(461\) −14.6490 −0.682272 −0.341136 0.940014i \(-0.610812\pi\)
−0.341136 + 0.940014i \(0.610812\pi\)
\(462\) 0 0
\(463\) 11.0973 0.515737 0.257869 0.966180i \(-0.416980\pi\)
0.257869 + 0.966180i \(0.416980\pi\)
\(464\) −2.99446 −0.139014
\(465\) −14.5727 −0.675793
\(466\) −11.9807 −0.554995
\(467\) 24.6923 1.14262 0.571311 0.820734i \(-0.306435\pi\)
0.571311 + 0.820734i \(0.306435\pi\)
\(468\) 29.3777 1.35798
\(469\) 0.0485769 0.00224307
\(470\) −5.14520 −0.237330
\(471\) 48.1276 2.21760
\(472\) 13.8292 0.636540
\(473\) 0 0
\(474\) 3.01726 0.138587
\(475\) 7.81494 0.358574
\(476\) −2.32986 −0.106789
\(477\) 3.03693 0.139051
\(478\) 6.46612 0.295753
\(479\) 31.2114 1.42608 0.713042 0.701121i \(-0.247317\pi\)
0.713042 + 0.701121i \(0.247317\pi\)
\(480\) −2.86564 −0.130798
\(481\) −7.81700 −0.356424
\(482\) −20.3441 −0.926646
\(483\) 22.8802 1.04109
\(484\) 0 0
\(485\) −5.33180 −0.242104
\(486\) 11.7710 0.533945
\(487\) 3.02777 0.137201 0.0686007 0.997644i \(-0.478147\pi\)
0.0686007 + 0.997644i \(0.478147\pi\)
\(488\) −7.82443 −0.354195
\(489\) −6.96466 −0.314953
\(490\) 1.00000 0.0451754
\(491\) 14.8069 0.668228 0.334114 0.942533i \(-0.391563\pi\)
0.334114 + 0.942533i \(0.391563\pi\)
\(492\) 19.7773 0.891630
\(493\) −6.97666 −0.314213
\(494\) 44.0504 1.98192
\(495\) 0 0
\(496\) 5.08533 0.228338
\(497\) −4.45361 −0.199772
\(498\) 31.0391 1.39089
\(499\) −6.99445 −0.313115 −0.156557 0.987669i \(-0.550040\pi\)
−0.156557 + 0.987669i \(0.550040\pi\)
\(500\) 1.00000 0.0447214
\(501\) −31.6096 −1.41221
\(502\) 12.5842 0.561663
\(503\) 1.93800 0.0864112 0.0432056 0.999066i \(-0.486243\pi\)
0.0432056 + 0.999066i \(0.486243\pi\)
\(504\) −5.21187 −0.232155
\(505\) 18.2701 0.813011
\(506\) 0 0
\(507\) −53.7947 −2.38910
\(508\) −6.58727 −0.292263
\(509\) 12.9518 0.574081 0.287040 0.957919i \(-0.407329\pi\)
0.287040 + 0.957919i \(0.407329\pi\)
\(510\) −6.67652 −0.295641
\(511\) −0.336814 −0.0148998
\(512\) 1.00000 0.0441942
\(513\) −49.5342 −2.18699
\(514\) 3.45995 0.152612
\(515\) −11.2347 −0.495061
\(516\) −33.0959 −1.45697
\(517\) 0 0
\(518\) 1.38681 0.0609327
\(519\) 42.1095 1.84840
\(520\) 5.63670 0.247185
\(521\) −5.90459 −0.258685 −0.129342 0.991600i \(-0.541287\pi\)
−0.129342 + 0.991600i \(0.541287\pi\)
\(522\) −15.6067 −0.683088
\(523\) 18.1944 0.795587 0.397794 0.917475i \(-0.369776\pi\)
0.397794 + 0.917475i \(0.369776\pi\)
\(524\) −1.20821 −0.0527808
\(525\) 2.86564 0.125067
\(526\) 8.01789 0.349597
\(527\) 11.8481 0.516111
\(528\) 0 0
\(529\) 40.7497 1.77173
\(530\) 0.582695 0.0253106
\(531\) 72.0759 3.12783
\(532\) −7.81494 −0.338821
\(533\) −38.9019 −1.68503
\(534\) 53.0912 2.29748
\(535\) −4.89157 −0.211481
\(536\) −0.0485769 −0.00209820
\(537\) −12.9353 −0.558198
\(538\) −18.5650 −0.800393
\(539\) 0 0
\(540\) −6.33840 −0.272761
\(541\) 15.1525 0.651457 0.325729 0.945463i \(-0.394390\pi\)
0.325729 + 0.945463i \(0.394390\pi\)
\(542\) −25.2051 −1.08265
\(543\) 29.1636 1.25153
\(544\) 2.32986 0.0998918
\(545\) −6.26442 −0.268338
\(546\) 16.1527 0.691272
\(547\) 10.3369 0.441975 0.220988 0.975277i \(-0.429072\pi\)
0.220988 + 0.975277i \(0.429072\pi\)
\(548\) 14.0829 0.601590
\(549\) −40.7799 −1.74044
\(550\) 0 0
\(551\) −23.4015 −0.996938
\(552\) −22.8802 −0.973846
\(553\) 1.05291 0.0447743
\(554\) 12.2281 0.519521
\(555\) 3.97408 0.168690
\(556\) −2.74290 −0.116325
\(557\) 6.41056 0.271624 0.135812 0.990735i \(-0.456636\pi\)
0.135812 + 0.990735i \(0.456636\pi\)
\(558\) 26.5041 1.12201
\(559\) 65.0996 2.75342
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −26.5934 −1.12178
\(563\) 36.3501 1.53197 0.765986 0.642857i \(-0.222251\pi\)
0.765986 + 0.642857i \(0.222251\pi\)
\(564\) 14.7443 0.620845
\(565\) 10.1651 0.427649
\(566\) 13.7220 0.576779
\(567\) −2.52795 −0.106164
\(568\) 4.45361 0.186869
\(569\) 45.1901 1.89447 0.947234 0.320544i \(-0.103866\pi\)
0.947234 + 0.320544i \(0.103866\pi\)
\(570\) −22.3948 −0.938014
\(571\) −24.7479 −1.03567 −0.517834 0.855481i \(-0.673262\pi\)
−0.517834 + 0.855481i \(0.673262\pi\)
\(572\) 0 0
\(573\) −34.3630 −1.43554
\(574\) 6.90154 0.288065
\(575\) 7.98434 0.332970
\(576\) 5.21187 0.217161
\(577\) −2.62695 −0.109361 −0.0546807 0.998504i \(-0.517414\pi\)
−0.0546807 + 0.998504i \(0.517414\pi\)
\(578\) −11.5718 −0.481322
\(579\) −51.2228 −2.12875
\(580\) −2.99446 −0.124338
\(581\) 10.8315 0.449366
\(582\) 15.2790 0.633334
\(583\) 0 0
\(584\) 0.336814 0.0139374
\(585\) 29.3777 1.21462
\(586\) −0.714463 −0.0295142
\(587\) 12.8971 0.532319 0.266159 0.963929i \(-0.414245\pi\)
0.266159 + 0.963929i \(0.414245\pi\)
\(588\) −2.86564 −0.118177
\(589\) 39.7416 1.63752
\(590\) 13.8292 0.569339
\(591\) −33.7775 −1.38942
\(592\) −1.38681 −0.0569973
\(593\) 33.4157 1.37222 0.686110 0.727498i \(-0.259317\pi\)
0.686110 + 0.727498i \(0.259317\pi\)
\(594\) 0 0
\(595\) −2.32986 −0.0955149
\(596\) 17.1642 0.703075
\(597\) 39.9303 1.63424
\(598\) 45.0053 1.84040
\(599\) 34.4823 1.40891 0.704455 0.709749i \(-0.251192\pi\)
0.704455 + 0.709749i \(0.251192\pi\)
\(600\) −2.86564 −0.116989
\(601\) 32.7460 1.33574 0.667869 0.744279i \(-0.267207\pi\)
0.667869 + 0.744279i \(0.267207\pi\)
\(602\) −11.5492 −0.470712
\(603\) −0.253176 −0.0103101
\(604\) −5.94304 −0.241819
\(605\) 0 0
\(606\) −52.3556 −2.12680
\(607\) 5.79559 0.235236 0.117618 0.993059i \(-0.462474\pi\)
0.117618 + 0.993059i \(0.462474\pi\)
\(608\) 7.81494 0.316938
\(609\) −8.58103 −0.347721
\(610\) −7.82443 −0.316802
\(611\) −29.0019 −1.17329
\(612\) 12.1429 0.490848
\(613\) 39.3606 1.58976 0.794879 0.606767i \(-0.207534\pi\)
0.794879 + 0.606767i \(0.207534\pi\)
\(614\) 19.3442 0.780667
\(615\) 19.7773 0.797498
\(616\) 0 0
\(617\) −16.3175 −0.656918 −0.328459 0.944518i \(-0.606529\pi\)
−0.328459 + 0.944518i \(0.606529\pi\)
\(618\) 32.1946 1.29506
\(619\) −44.7221 −1.79753 −0.898767 0.438427i \(-0.855536\pi\)
−0.898767 + 0.438427i \(0.855536\pi\)
\(620\) 5.08533 0.204232
\(621\) −50.6079 −2.03083
\(622\) 1.82729 0.0732675
\(623\) 18.5268 0.742263
\(624\) −16.1527 −0.646626
\(625\) 1.00000 0.0400000
\(626\) −9.83108 −0.392929
\(627\) 0 0
\(628\) −16.7947 −0.670183
\(629\) −3.23106 −0.128831
\(630\) −5.21187 −0.207646
\(631\) −30.1105 −1.19868 −0.599340 0.800495i \(-0.704570\pi\)
−0.599340 + 0.800495i \(0.704570\pi\)
\(632\) −1.05291 −0.0418825
\(633\) 75.7951 3.01258
\(634\) −13.8545 −0.550232
\(635\) −6.58727 −0.261408
\(636\) −1.66979 −0.0662115
\(637\) 5.63670 0.223334
\(638\) 0 0
\(639\) 23.2116 0.918237
\(640\) 1.00000 0.0395285
\(641\) −0.937781 −0.0370401 −0.0185201 0.999828i \(-0.505895\pi\)
−0.0185201 + 0.999828i \(0.505895\pi\)
\(642\) 14.0175 0.553225
\(643\) 44.9904 1.77425 0.887125 0.461529i \(-0.152699\pi\)
0.887125 + 0.461529i \(0.152699\pi\)
\(644\) −7.98434 −0.314627
\(645\) −33.0959 −1.30315
\(646\) 18.2077 0.716373
\(647\) −32.9160 −1.29406 −0.647032 0.762463i \(-0.723990\pi\)
−0.647032 + 0.762463i \(0.723990\pi\)
\(648\) 2.52795 0.0993072
\(649\) 0 0
\(650\) 5.63670 0.221089
\(651\) 14.5727 0.571149
\(652\) 2.43041 0.0951821
\(653\) −3.32100 −0.129961 −0.0649804 0.997887i \(-0.520698\pi\)
−0.0649804 + 0.997887i \(0.520698\pi\)
\(654\) 17.9516 0.701961
\(655\) −1.20821 −0.0472086
\(656\) −6.90154 −0.269460
\(657\) 1.75543 0.0684857
\(658\) 5.14520 0.200581
\(659\) 20.5642 0.801069 0.400535 0.916282i \(-0.368824\pi\)
0.400535 + 0.916282i \(0.368824\pi\)
\(660\) 0 0
\(661\) −43.9387 −1.70902 −0.854508 0.519438i \(-0.826141\pi\)
−0.854508 + 0.519438i \(0.826141\pi\)
\(662\) 8.02084 0.311739
\(663\) −37.6335 −1.46157
\(664\) −10.8315 −0.420343
\(665\) −7.81494 −0.303050
\(666\) −7.22784 −0.280073
\(667\) −23.9088 −0.925752
\(668\) 11.0306 0.426786
\(669\) −70.4640 −2.72430
\(670\) −0.0485769 −0.00187669
\(671\) 0 0
\(672\) 2.86564 0.110544
\(673\) 0.623385 0.0240297 0.0120149 0.999928i \(-0.496175\pi\)
0.0120149 + 0.999928i \(0.496175\pi\)
\(674\) 11.6060 0.447045
\(675\) −6.33840 −0.243965
\(676\) 18.7723 0.722013
\(677\) 30.6611 1.17840 0.589200 0.807987i \(-0.299443\pi\)
0.589200 + 0.807987i \(0.299443\pi\)
\(678\) −29.1294 −1.11871
\(679\) 5.33180 0.204616
\(680\) 2.32986 0.0893460
\(681\) −16.0258 −0.614112
\(682\) 0 0
\(683\) −4.08570 −0.156335 −0.0781676 0.996940i \(-0.524907\pi\)
−0.0781676 + 0.996940i \(0.524907\pi\)
\(684\) 40.7304 1.55737
\(685\) 14.0829 0.538079
\(686\) −1.00000 −0.0381802
\(687\) 23.2214 0.885951
\(688\) 11.5492 0.440311
\(689\) 3.28447 0.125128
\(690\) −22.8802 −0.871035
\(691\) −9.75385 −0.371054 −0.185527 0.982639i \(-0.559399\pi\)
−0.185527 + 0.982639i \(0.559399\pi\)
\(692\) −14.6947 −0.558607
\(693\) 0 0
\(694\) 26.4301 1.00327
\(695\) −2.74290 −0.104044
\(696\) 8.58103 0.325263
\(697\) −16.0796 −0.609059
\(698\) −27.3777 −1.03626
\(699\) 34.3323 1.29857
\(700\) −1.00000 −0.0377964
\(701\) 8.34928 0.315348 0.157674 0.987491i \(-0.449600\pi\)
0.157674 + 0.987491i \(0.449600\pi\)
\(702\) −35.7276 −1.34845
\(703\) −10.8378 −0.408755
\(704\) 0 0
\(705\) 14.7443 0.555301
\(706\) 21.3772 0.804541
\(707\) −18.2701 −0.687119
\(708\) −39.6294 −1.48937
\(709\) 41.6502 1.56421 0.782103 0.623149i \(-0.214147\pi\)
0.782103 + 0.623149i \(0.214147\pi\)
\(710\) 4.45361 0.167141
\(711\) −5.48763 −0.205802
\(712\) −18.5268 −0.694323
\(713\) 40.6030 1.52059
\(714\) 6.67652 0.249863
\(715\) 0 0
\(716\) 4.51392 0.168693
\(717\) −18.5295 −0.691998
\(718\) 10.8904 0.406427
\(719\) 27.1930 1.01413 0.507064 0.861909i \(-0.330731\pi\)
0.507064 + 0.861909i \(0.330731\pi\)
\(720\) 5.21187 0.194235
\(721\) 11.2347 0.418403
\(722\) 42.0733 1.56581
\(723\) 58.2986 2.16815
\(724\) −10.1770 −0.378225
\(725\) −2.99446 −0.111211
\(726\) 0 0
\(727\) 26.4443 0.980766 0.490383 0.871507i \(-0.336857\pi\)
0.490383 + 0.871507i \(0.336857\pi\)
\(728\) −5.63670 −0.208910
\(729\) −41.3153 −1.53020
\(730\) 0.336814 0.0124660
\(731\) 26.9081 0.995232
\(732\) 22.4220 0.828740
\(733\) −25.1965 −0.930653 −0.465326 0.885139i \(-0.654063\pi\)
−0.465326 + 0.885139i \(0.654063\pi\)
\(734\) 11.3108 0.417487
\(735\) −2.86564 −0.105701
\(736\) 7.98434 0.294307
\(737\) 0 0
\(738\) −35.9699 −1.32407
\(739\) −5.90682 −0.217286 −0.108643 0.994081i \(-0.534651\pi\)
−0.108643 + 0.994081i \(0.534651\pi\)
\(740\) −1.38681 −0.0509800
\(741\) −126.233 −4.63727
\(742\) −0.582695 −0.0213914
\(743\) −19.5944 −0.718847 −0.359424 0.933175i \(-0.617027\pi\)
−0.359424 + 0.933175i \(0.617027\pi\)
\(744\) −14.5727 −0.534261
\(745\) 17.1642 0.628849
\(746\) −11.6658 −0.427116
\(747\) −56.4523 −2.06548
\(748\) 0 0
\(749\) 4.89157 0.178734
\(750\) −2.86564 −0.104638
\(751\) 0.465285 0.0169785 0.00848925 0.999964i \(-0.497298\pi\)
0.00848925 + 0.999964i \(0.497298\pi\)
\(752\) −5.14520 −0.187626
\(753\) −36.0619 −1.31417
\(754\) −16.8789 −0.614692
\(755\) −5.94304 −0.216289
\(756\) 6.33840 0.230525
\(757\) −6.85681 −0.249215 −0.124607 0.992206i \(-0.539767\pi\)
−0.124607 + 0.992206i \(0.539767\pi\)
\(758\) 0.617088 0.0224136
\(759\) 0 0
\(760\) 7.81494 0.283478
\(761\) −41.5119 −1.50481 −0.752403 0.658703i \(-0.771105\pi\)
−0.752403 + 0.658703i \(0.771105\pi\)
\(762\) 18.8767 0.683831
\(763\) 6.26442 0.226787
\(764\) 11.9914 0.433834
\(765\) 12.1429 0.439028
\(766\) −26.4471 −0.955573
\(767\) 77.9510 2.81465
\(768\) −2.86564 −0.103405
\(769\) 17.8632 0.644164 0.322082 0.946712i \(-0.395617\pi\)
0.322082 + 0.946712i \(0.395617\pi\)
\(770\) 0 0
\(771\) −9.91495 −0.357078
\(772\) 17.8748 0.643330
\(773\) 9.91463 0.356604 0.178302 0.983976i \(-0.442940\pi\)
0.178302 + 0.983976i \(0.442940\pi\)
\(774\) 60.1931 2.16360
\(775\) 5.08533 0.182670
\(776\) −5.33180 −0.191400
\(777\) −3.97408 −0.142569
\(778\) 1.60446 0.0575226
\(779\) −53.9352 −1.93243
\(780\) −16.1527 −0.578360
\(781\) 0 0
\(782\) 18.6024 0.665220
\(783\) 18.9801 0.678293
\(784\) 1.00000 0.0357143
\(785\) −16.7947 −0.599430
\(786\) 3.46228 0.123496
\(787\) 0.812618 0.0289667 0.0144834 0.999895i \(-0.495390\pi\)
0.0144834 + 0.999895i \(0.495390\pi\)
\(788\) 11.7871 0.419897
\(789\) −22.9763 −0.817979
\(790\) −1.05291 −0.0374609
\(791\) −10.1651 −0.361429
\(792\) 0 0
\(793\) −44.1039 −1.56618
\(794\) 5.28819 0.187671
\(795\) −1.66979 −0.0592214
\(796\) −13.9342 −0.493884
\(797\) 32.3331 1.14530 0.572649 0.819800i \(-0.305916\pi\)
0.572649 + 0.819800i \(0.305916\pi\)
\(798\) 22.3948 0.792766
\(799\) −11.9876 −0.424090
\(800\) 1.00000 0.0353553
\(801\) −96.5595 −3.41176
\(802\) −13.6059 −0.480441
\(803\) 0 0
\(804\) 0.139204 0.00490934
\(805\) −7.98434 −0.281411
\(806\) 28.6645 1.00966
\(807\) 53.2005 1.87275
\(808\) 18.2701 0.642741
\(809\) 24.5501 0.863135 0.431568 0.902081i \(-0.357961\pi\)
0.431568 + 0.902081i \(0.357961\pi\)
\(810\) 2.52795 0.0888230
\(811\) −41.3394 −1.45162 −0.725812 0.687893i \(-0.758536\pi\)
−0.725812 + 0.687893i \(0.758536\pi\)
\(812\) 2.99446 0.105085
\(813\) 72.2287 2.53317
\(814\) 0 0
\(815\) 2.43041 0.0851335
\(816\) −6.67652 −0.233725
\(817\) 90.2567 3.15768
\(818\) 3.40230 0.118959
\(819\) −29.3777 −1.02654
\(820\) −6.90154 −0.241012
\(821\) −16.0197 −0.559090 −0.279545 0.960133i \(-0.590184\pi\)
−0.279545 + 0.960133i \(0.590184\pi\)
\(822\) −40.3564 −1.40759
\(823\) 19.3583 0.674787 0.337394 0.941364i \(-0.390455\pi\)
0.337394 + 0.941364i \(0.390455\pi\)
\(824\) −11.2347 −0.391380
\(825\) 0 0
\(826\) −13.8292 −0.481179
\(827\) −43.1757 −1.50136 −0.750682 0.660663i \(-0.770275\pi\)
−0.750682 + 0.660663i \(0.770275\pi\)
\(828\) 41.6133 1.44616
\(829\) −38.5839 −1.34007 −0.670037 0.742327i \(-0.733722\pi\)
−0.670037 + 0.742327i \(0.733722\pi\)
\(830\) −10.8315 −0.375966
\(831\) −35.0412 −1.21557
\(832\) 5.63670 0.195417
\(833\) 2.32986 0.0807248
\(834\) 7.86016 0.272175
\(835\) 11.0306 0.381729
\(836\) 0 0
\(837\) −32.2329 −1.11413
\(838\) −39.5263 −1.36541
\(839\) 9.31984 0.321757 0.160878 0.986974i \(-0.448567\pi\)
0.160878 + 0.986974i \(0.448567\pi\)
\(840\) 2.86564 0.0988738
\(841\) −20.0332 −0.690800
\(842\) −15.6617 −0.539738
\(843\) 76.2071 2.62471
\(844\) −26.4497 −0.910435
\(845\) 18.7723 0.645788
\(846\) −26.8161 −0.921956
\(847\) 0 0
\(848\) 0.582695 0.0200098
\(849\) −39.3223 −1.34954
\(850\) 2.32986 0.0799135
\(851\) −11.0727 −0.379568
\(852\) −12.7624 −0.437233
\(853\) −9.65688 −0.330645 −0.165323 0.986240i \(-0.552867\pi\)
−0.165323 + 0.986240i \(0.552867\pi\)
\(854\) 7.82443 0.267747
\(855\) 40.7304 1.39295
\(856\) −4.89157 −0.167190
\(857\) −7.88344 −0.269293 −0.134647 0.990894i \(-0.542990\pi\)
−0.134647 + 0.990894i \(0.542990\pi\)
\(858\) 0 0
\(859\) 9.00176 0.307136 0.153568 0.988138i \(-0.450924\pi\)
0.153568 + 0.988138i \(0.450924\pi\)
\(860\) 11.5492 0.393826
\(861\) −19.7773 −0.674009
\(862\) −5.63150 −0.191810
\(863\) −14.5890 −0.496616 −0.248308 0.968681i \(-0.579874\pi\)
−0.248308 + 0.968681i \(0.579874\pi\)
\(864\) −6.33840 −0.215637
\(865\) −14.6947 −0.499633
\(866\) −34.5223 −1.17311
\(867\) 33.1604 1.12619
\(868\) −5.08533 −0.172607
\(869\) 0 0
\(870\) 8.58103 0.290924
\(871\) −0.273813 −0.00927781
\(872\) −6.26442 −0.212140
\(873\) −27.7886 −0.940502
\(874\) 62.3972 2.11062
\(875\) −1.00000 −0.0338062
\(876\) −0.965185 −0.0326106
\(877\) −17.0812 −0.576791 −0.288396 0.957511i \(-0.593122\pi\)
−0.288396 + 0.957511i \(0.593122\pi\)
\(878\) −30.0533 −1.01425
\(879\) 2.04739 0.0690568
\(880\) 0 0
\(881\) −30.9541 −1.04287 −0.521435 0.853291i \(-0.674603\pi\)
−0.521435 + 0.853291i \(0.674603\pi\)
\(882\) 5.21187 0.175493
\(883\) −13.5966 −0.457561 −0.228781 0.973478i \(-0.573474\pi\)
−0.228781 + 0.973478i \(0.573474\pi\)
\(884\) 13.1327 0.441700
\(885\) −39.6294 −1.33213
\(886\) −32.0651 −1.07725
\(887\) −17.1901 −0.577188 −0.288594 0.957452i \(-0.593188\pi\)
−0.288594 + 0.957452i \(0.593188\pi\)
\(888\) 3.97408 0.133361
\(889\) 6.58727 0.220930
\(890\) −18.5268 −0.621021
\(891\) 0 0
\(892\) 24.5893 0.823311
\(893\) −40.2094 −1.34556
\(894\) −49.1865 −1.64504
\(895\) 4.51392 0.150884
\(896\) −1.00000 −0.0334077
\(897\) −128.969 −4.30614
\(898\) −9.27706 −0.309579
\(899\) −15.2278 −0.507876
\(900\) 5.21187 0.173729
\(901\) 1.35760 0.0452281
\(902\) 0 0
\(903\) 33.0959 1.10136
\(904\) 10.1651 0.338086
\(905\) −10.1770 −0.338295
\(906\) 17.0306 0.565803
\(907\) −23.5921 −0.783363 −0.391682 0.920101i \(-0.628107\pi\)
−0.391682 + 0.920101i \(0.628107\pi\)
\(908\) 5.59242 0.185591
\(909\) 95.2215 3.15830
\(910\) −5.63670 −0.186855
\(911\) 50.0697 1.65888 0.829441 0.558594i \(-0.188659\pi\)
0.829441 + 0.558594i \(0.188659\pi\)
\(912\) −22.3948 −0.741565
\(913\) 0 0
\(914\) 20.0782 0.664127
\(915\) 22.4220 0.741247
\(916\) −8.10340 −0.267744
\(917\) 1.20821 0.0398985
\(918\) −14.7676 −0.487403
\(919\) 13.1628 0.434200 0.217100 0.976149i \(-0.430340\pi\)
0.217100 + 0.976149i \(0.430340\pi\)
\(920\) 7.98434 0.263236
\(921\) −55.4333 −1.82659
\(922\) −14.6490 −0.482439
\(923\) 25.1036 0.826296
\(924\) 0 0
\(925\) −1.38681 −0.0455979
\(926\) 11.0973 0.364681
\(927\) −58.5539 −1.92316
\(928\) −2.99446 −0.0982979
\(929\) −9.87262 −0.323910 −0.161955 0.986798i \(-0.551780\pi\)
−0.161955 + 0.986798i \(0.551780\pi\)
\(930\) −14.5727 −0.477858
\(931\) 7.81494 0.256124
\(932\) −11.9807 −0.392441
\(933\) −5.23634 −0.171430
\(934\) 24.6923 0.807956
\(935\) 0 0
\(936\) 29.3777 0.960240
\(937\) −4.81983 −0.157457 −0.0787285 0.996896i \(-0.525086\pi\)
−0.0787285 + 0.996896i \(0.525086\pi\)
\(938\) 0.0485769 0.00158609
\(939\) 28.1723 0.919368
\(940\) −5.14520 −0.167818
\(941\) 57.3904 1.87087 0.935437 0.353493i \(-0.115006\pi\)
0.935437 + 0.353493i \(0.115006\pi\)
\(942\) 48.1276 1.56808
\(943\) −55.1043 −1.79444
\(944\) 13.8292 0.450102
\(945\) 6.33840 0.206188
\(946\) 0 0
\(947\) −4.22221 −0.137203 −0.0686017 0.997644i \(-0.521854\pi\)
−0.0686017 + 0.997644i \(0.521854\pi\)
\(948\) 3.01726 0.0979959
\(949\) 1.89852 0.0616284
\(950\) 7.81494 0.253550
\(951\) 39.7019 1.28742
\(952\) −2.32986 −0.0755111
\(953\) 5.10500 0.165367 0.0826836 0.996576i \(-0.473651\pi\)
0.0826836 + 0.996576i \(0.473651\pi\)
\(954\) 3.03693 0.0983241
\(955\) 11.9914 0.388033
\(956\) 6.46612 0.209129
\(957\) 0 0
\(958\) 31.2114 1.00839
\(959\) −14.0829 −0.454759
\(960\) −2.86564 −0.0924880
\(961\) −5.13941 −0.165787
\(962\) −7.81700 −0.252030
\(963\) −25.4942 −0.821539
\(964\) −20.3441 −0.655238
\(965\) 17.8748 0.575412
\(966\) 22.8802 0.736159
\(967\) −61.6521 −1.98260 −0.991298 0.131633i \(-0.957978\pi\)
−0.991298 + 0.131633i \(0.957978\pi\)
\(968\) 0 0
\(969\) −52.1766 −1.67616
\(970\) −5.33180 −0.171194
\(971\) −26.0709 −0.836656 −0.418328 0.908296i \(-0.637384\pi\)
−0.418328 + 0.908296i \(0.637384\pi\)
\(972\) 11.7710 0.377556
\(973\) 2.74290 0.0879334
\(974\) 3.02777 0.0970160
\(975\) −16.1527 −0.517301
\(976\) −7.82443 −0.250454
\(977\) 19.0437 0.609261 0.304631 0.952471i \(-0.401467\pi\)
0.304631 + 0.952471i \(0.401467\pi\)
\(978\) −6.96466 −0.222705
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −32.6493 −1.04241
\(982\) 14.8069 0.472509
\(983\) −19.3347 −0.616681 −0.308340 0.951276i \(-0.599774\pi\)
−0.308340 + 0.951276i \(0.599774\pi\)
\(984\) 19.7773 0.630478
\(985\) 11.7871 0.375568
\(986\) −6.97666 −0.222182
\(987\) −14.7443 −0.469315
\(988\) 44.0504 1.40143
\(989\) 92.2131 2.93221
\(990\) 0 0
\(991\) −28.4567 −0.903958 −0.451979 0.892029i \(-0.649282\pi\)
−0.451979 + 0.892029i \(0.649282\pi\)
\(992\) 5.08533 0.161459
\(993\) −22.9848 −0.729401
\(994\) −4.45361 −0.141260
\(995\) −13.9342 −0.441743
\(996\) 31.0391 0.983511
\(997\) −14.1144 −0.447008 −0.223504 0.974703i \(-0.571750\pi\)
−0.223504 + 0.974703i \(0.571750\pi\)
\(998\) −6.99445 −0.221406
\(999\) 8.79013 0.278107
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8470.2.a.dc.1.1 6
11.7 odd 10 770.2.n.j.71.3 12
11.8 odd 10 770.2.n.j.141.3 yes 12
11.10 odd 2 8470.2.a.cw.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.j.71.3 12 11.7 odd 10
770.2.n.j.141.3 yes 12 11.8 odd 10
8470.2.a.cw.1.1 6 11.10 odd 2
8470.2.a.dc.1.1 6 1.1 even 1 trivial